Question

Divide the following polynomial by synthetic division method and also by linear division method. Write the quotient and the remainder.

`(x^4 - 3x^2 - 8) ÷ (x + 4)`

Answer 1

Synthetic Division:

Dividend = `x^4 - 3x^2 - 8 = x^4 +0x^3 - 3x^2 + 0x - 8`

Divisor = x + 4

Opposite of 4 = −4

The coefficient form of the quotient is (1, −4, 13, −52). 

∴ Quotient = x³ − 4x² + 13x − 52 and Remainder = 200 

Linear Method:

`x^4 - 3x^2 - 8`

`= x^3 (x + 4) - 4x^3 - 3x^2 - 8`

`= x^3 (x + 4) - 4x^2 (x +4) + 16x^2 -3x^2 - 8`

`= x^3 (x + 4) - 4x^2 (x + 4) + 13x (x +4) - 52x - 8`

`= x^3 (x + 4) - 4x^2 (x + 4) + 13x (x +4) - 52(x + 4) + 208 - 8`

`= (x + 4) (x^3 - 4x^2 + 13 x - 52) + 200`